Growth and Self-Assembly of CsPbBr3 Nanocrystals in the TOPO/PbBr2 Synthesis as Seen with X-ray Scattering

Despite broad interest in colloidal lead halide perovskite nanocrystals (LHP NCs), their intrinsic fast growth has prevented controlled synthesis of small, monodisperse crystals and insights into the reaction mechanism. Recently, a much slower synthesis of LHP NCs with extreme size control has been reported, based on diluted TOPO/PbBr2 precursors and a diisooctylphosphinate capping ligand. We report new insights into the nucleation, growth, and self-assembly in this reaction, obtained by in situ synchrotron-based small-angle X-ray scattering and optical absorption spectroscopy. We show that dispersed 3 nm Cs[PbBr3] agglomerates are the key intermediate species: first, they slowly nucleate into crystals, and then they release Cs[PbBr3] monomers for further growth of the crystals. We show the merits of a low Cs[PbBr3] monomer concentration for the reaction based on oleate ligands. We also examine the spontaneous superlattice formation mechanism occurring when the growing nanocrystals in the solvent reach a critical size of 11.6 nm.

In situ X-ray scattering measurements

Data acquisition
The SAXS/WAXS experiment was performed at beamline P21.2 at Petra III (DESY) synchrotron in Hamburg.
We used an energy of 37.5 keV (0.0331 nm) with VAREX XRD4343CT detector and a sample-to-detector distance of 1.5 m (for WAXS) and 15 m (SAXS). These distances allowed us to probe a q range between 16 nm -1 and 58 nm -1 for the WAXS and between 0.15 nm -1 and 2.2 nm -1 for the SAXS. A LaB6 standard was used for calibration of the q range, of the scattering intensity and of the instrumental resolution, while absolute intensities were obtained by calibrating with glassy carbon and normalizing for the path length. 2 The acquisition time was typically 0.5 s or 2 s, depending on the dilution of the precursors.

Analysis of the SAXS data
For the analysis of the SAXS data, we modeled the scattering pattern of a dispersion of nanocrystals with an isotropic form factor and a Gaussian distribution. Based on former reports, 3 the shape of the nanocrystals is quasi-spherical (in contrast to conventional cesium LHP nanocrystals 4 ), hence our choice to use a spherical form factor to describe our scattering objects. This assumption is also supported by the results of the fitting of the scattering curve with a shape-retrieval algorithm based on dummy atoms (Fig. 1b).
The scattering intensity is therefore expressed by the formula 5 : is the molar concentration of nanocrystals, is the Avogadro number and ℎ ( ) is the isotropic scattering form factor of a spherical scattering object, which is given by: If we also implement the Gaussian distribution of radii R, the form factor will assume the form of: Where 0 is the average radius of the distribution with standard deviation and ∆ is the scattering contrast, defined as the difference in the scattering length density of the nanocrystals and the solvent: The scattering length densities can be calculated with the following equation: Where λ is the wavelength of the X-ray photons (0.331 Å) and δ is the real part of the refractive index of the material, which is expressed as = 1 + − . Since the imaginary part of the refractive index β is orders of magnitude smaller than the real part (since we do not work in the vicinity of an X-ray absorption edge), we For the analysis of the SAXS patterns after injection of the precursors, the SAXS pattern acquired before the injection has been used as background. In Figure S2 we show one of the typical SAXS patterns acquired for our experiments before precursor injection. All the SAXS patterns acquired before precursor injection show the complete absence of any scattering object.

Estimation of error on experimental geometry
Due to the thickness of the probed sample (1 cm diameter vial), assumptions for a punctiform scattering object are debatable. Therefore, we estimated the effect of the spatial aberration on the signal acquired by the WAXS and SAXS detectors. Considering the geometry of Figure S1, the sample-to-detector distance is 150 cm ( ̅̅̅̅ ) and 151 cm ( ̅̅̅̅ ) respectively for the two ends of the sample (1 cm wide). The spatial aberration on the detector, ̅̅̅̅ , can then be described as: Considering the two extremes of β in our experiment, determined by the physical position of the WAXS detector, as 8° ( ̅̅̅̅ and ̅̅̅̅ ) and 25° ( ̅̅̅̅ ), the spatial aberration would be 0.14 cm and 0.42 cm respectively.
Considering that each pixel is ~0.15 cm, the total error associated to spatial aberration at the two extremes of the detector will be between 0.1° and 0.3° respectively for 8° and 25°. Such small error implies that the assumption of a punctiform scattering object is still valid. The larger distance of the SAXS detector (15 m) leads to an even smaller error for this measurement.

Reaction yield calculation
After obtaining the molar concentration and size distribution of the CsPbBr3 nanocrystals from the fitting of the scattering data, we extracted the average number of Cs + cations + in a nanocrystal from the ratio between the average nanocrystals volume and the volume of the CsPbBr3 unit cell (0.602 nm): The reaction yield (RY) is then defined from the ratio between the total amount of Cs + cations incorporated in the nanocrystals and the amount of Cs + cation present in solution at the beginning of the reaction: Here, , is the total concentration of nanocrystals, as extracted from the SAXS data analysis, and is the initial molar concentration of the Cs-precursor.

Analysis of the self-assembled crystalline structure
In order to determine the crystal structure in which the nanocrystals self-assemble in solution, we scanned through different crystal structures in order of decreasing symmetry (i.e. cubic, hexagonal, tetragonal, orthorhombic). For each crystal structure we fitted the experimental peak positions to the allowed reflections with a weighted least square fitting procedure in a custom script, the variable being the lattice parameters. The optimal result is obtained by fitting with a face-centered orthorhombic crystal structure with lattice parameters: 14.6, 16.0 and 17.8 nm. The position of each allowed reflection is calculated as: With h, k, l as Miller indices of the allowed reflections (i.e. reflections for which h, k, l are all odd or even) and a, b, c the lattice parameters.
The structure factor of the superstructures ( , ) was obtained by dividing each SAXS pattern ( , ) by the effective form factor ( , ), i.e. the scattering pattern of non-interacting nanocrystals before self-assembly.
After indexing each experimental peak of the scattering pattern, we could extract the average NC-NC distance along the {002} direction by following the evolution of the position of the {002} peak (PP{002}):

Particle reconstruction
Particle reconstruction models were obtained by independently fitting the scattering curves with a commercially available algorithm based on dummy atoms (SasHel). 6

NMR analysis
Solution 31 P NMR spectra were recorded on a Bruker 11.7 T spectrometer equipped with an AVANCE III console and a PABBO probe. A 30° excitation pulse (4.7 µs) and 1 H decoupling were used. 31 P chemical shifts were referenced externally relative to 85% H3PO4 in H2O.

In situ optical absorbance measurements
In situ optical absorbance measurements were performed using a custom-made three-neck flask 7 equipped with an indentation, allowing to probe a small path length (1 mm) when performing the synthesis in high concentrations. The absorbance of the crude mixture was probed with an Ocean Optics deuterium-tungsten light source (DH-2000-BAL-TTL-24V) and an Ocean Optics OCEAN-HDX-XR spectrometer.

Ex situ optical absorbance measurements
Ex situ optical absorbance measurements were performed using a Jasco V670 spectrometer in transmission mode. The sample was injected in a 150 µm gap between two glued glass slides.       Compared to the synthesis in Fig. 3 of the main text, this synthesis was 2.5x more diluted. The evolution of the first excitonic peak is more evident here thanks to the lower particle concentration.     The synthetic conditions are in all similar to the ones used for the data presented in Figure 2 of the main text, with the only difference that we used Cs-OA as Cs precursor instead of Cs-DOPA. We remark the striking difference to the absorption spectra of Figure 2: the appearance of excitonic features associated to NCs is here faster than the time resolution, pointing towards a fast (< 250 ms) nucleation event.   the only difference being the Cs precursor: Cs-DOPA (blue) and Cs-OA (red). When the NCs are stabilized by Cs-DOPA they start to spontaneously self-assemble into bigger superstructures at the critical size of 11.6 nm. In contrast, when they are stabilized by Cs-OA, this behavior is not observed, despite growing to even bigger sizes.